Generalized Equations and Their Solutions in the (S, 0) (0, S) Representations of the Lorentz Group

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1 Journal of Physics: Conference Series PAPER OPEN ACCESS Generalized Equations and Their Solutions in the (S, 0 (0, S Representations of the Lorentz Group To cite this article: V V Dvoeglazov 2017 J. Phys.: Conf. Ser Related content - Generalized equations of state and regular universes F Contreras, N Cruz and E González - Squeezed states and uncertainty relations Octavio Castaños, Rocio Jauregue- Renaud, Young S Kim et al. - Majorana Neutrino: Chirality and Helicity V V Dvoeglazov View the article online for updates and enhancements. This content was downloaded from IP address on 26/12/2017 at 16:57

2 Generalized Equations and Their Solutions in the (S,0 (0,S Representations of the Lorentz Group V V Dvoeglazov UAF, Universidad Autónoma de Zacatecas Apartado Postal 636, Zacatecas Zac., México valeri@fisica.uaz.edu.mx Abstract. We present three explicit examples of generalizations in relativistic quantum mechanics. First of all, we discuss the generalized spin-1/2 equations for neutrinos. They have been obtained by means of the Gersten-Sakurai method for derivations of arbitrary-spin relativistic equations. Possible physical consequences are discussed. Next, it is easy to check that both Dirac algebraic equation Det(ˆp m = 0 and Det(ˆp + m = 0 for u and v 4-spinors have solutions with p 0 = ±E p = ± p 2 +m 2. The same is true for higher-spin equations. Meanwhile, every book considers the equality p 0 = E p for both u and v spinors of the (1/2,0 (0,1/2 representation only, thus applying the Dirac-Feynman-Stueckelberg procedure for elimination of the negative-energy solutions. The recent Ziino works (and, independently, the articles of several others show that the Fock space can be doubled. We re-consider this possibility on the quantum field level for both S = 1/2 and higher spin particles. The third example is: we postulate the non-commutativity of 4-momenta, and we derive the mass splitting in the Dirac equation. Some applications are discussed. 1. Generalized neutrino equations A. Gersten[1] proposed a method for derivations of massless equations of arbitrary-spin particles. In fact, his method is related to the van der Waerden-Sakurai [2] procedure for the derivation of the massive Dirac equation. I commented the derivation of the Maxwell equations in [3]. Then, I showed that the method is rather ambigious, because instead of free-space Maxwell equations one can obtain generalized S = 1 equations, which connect the antisymmetric tensor field with additional scalar fields. The problem of physical significance of additional scalar chi-fields should be solved, of course, by experiment. In the present contribution we apply the van der Waerden-Sakurai-Gersten procedure to the spin-1/2 fields. As a result we obtain equations which generalize the well-known Weyl equations. However, these equations are known for a long time [4]. Raspini [5,6] analized them again in detail. I add some comments on physical contents of the generalized spin-1/2 equations. I use the equation (4 of the Gersten paper [1a] for the two-component spinor field function: [ ][ ] (E 2 c 2 p 2 I (2 ψ = EI (2 c p σ EI (2 +c p σ ψ = 0. (1 The matrix I (2 is the identity matrix of order 2; σ is the vector composed of the Pauli matrices; c is the light speed, E and p form the energy-momentum vector. Actually, this equation is Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1

3 the massless limit of the equation which has been presented (together with the corresponding method of derivation of the Dirac equation in the Sakurai book [2]. In the latter case one should substitute m 2 c 4 into the right-hand side of Eq. (1. However, instead of the equation (3.25 of [2] one can define ad hoc the two-component right field function φ R = 1 m 1 c (i h x 0 i hσ ψ, φ L = ψ (2 with the different mass parameter m 1. In such a way we come to the system of the first-order differential equations It can be re-written in the 4-component form: (i h +i hσ φ R = m2 2 c φ L, x 0 m 1 (3 (i h i hσ φ L = m 1 cφ R. x 0 (4 ( ( i h( / x0 i hσ ψa = (5 i hσ i h( / x 0 ψ B = c ( (m 2 2 /m 1 +m 1 ( m 2 2 /m ( 1 +m 1 ψa 2 ( m 2 2 /m 1 +m 1 (m 2 2 /m 1 +m 1 for the function Ψ = column(ψ A ψ B = column(φ R +φ L φ R φ L. The generalized equation (5 can be written in the covariant form. [ iγ µ µ m2 2 c m 1 h (1 γ 5 2 m 1c h ψ B (1+γ 5 ] Ψ = 0. (6 2 The standard representation of γ µ matrices has been used here. If m 1 = m 2 we can recover the standard Dirac equation. As noted in [4b] this procedure can be viewed as the simple change of the representation of γ µ matrices. However, this is valid unless m 2 0 only. Otherwise, the entries in the transformation matrix become to be singular. Furthermore, one can either repeat a similar procedure (the modified Sakurai procedure starting from the massless equation (4 of [1a] or put m 2 = 0 in eq. (6. The massless equation is 1 [ iγ µ µ m 1c h (1+γ 5 ] Ψ = 0. (7 2 Then, we may have different physical consequences following from (7 comparing with those which follow from the Weyl equation. 2 The mathematical reason of such a possibility of different massless limits is that the corresponding change of representation of γ µ matrices involves mass parameters m 1 and m 2 themselves. It is interesting to note that we can also repeat this procedure for other ad hoc definition (or for even more general definitions; φ L = 1 m 3 c (i h x 0 +i hσ ψ, φ R = ψ. (8 1 It is necesary to stress that the term massless is used in the sense that p µp µ = 0. 2 Remember that the Weyl equation is obtained as m 0 limit of the usual Dirac equation. 2

4 This is due to the fact that the parity properties of the two-component spinor are undefined in the two-component equation. The resulting equation is [ iγ µ µ m2 4 c (1+γ 5 m 3c(1 γ 5 ] Ψ = 0, (9 m 3 h 2 h 2 which gives us yet another equation in the massless limit (m 4 0: [ iγ µ µ m 3c(1 γ 5 ] Ψ = 0, (10 h 2 differing in the sign at the γ 5 term from (7. The above procedure can be generalized to any Lorentz group representations, i. e., to any spins. Is the physical content of the generalized S = 1/2 massless equations the same as that of the Weyl equation? Our answer is no. The excellent discussion can be found in [4a,b]. First of all, the theory does not have chiral invariance. Those authors call the additional parameters as the measures of the degree of chirality. Apart of this, Tokuoka introduced the concept of the gauge transformations (not to confuse with phase transformations for the 4-spinor fields. He also found some strange properties of the anti-commutation relations(see 3 in[4a]. And finally, the equation (7 describes four states, two of which answer for the positive energy p 0 = p, and two others answer for the negative energy p 0 = p. I just want to add the following speculations to the discussion. The operator of the chiralhelicity ˆη = (α ˆp (in the spinorial representation used in [4b] does not commute, e.g., with the Hamiltonian of the equation (7: 3 [H,α ˆp] = 2 m 1c h 1 γ 5 (γ ˆp. (11 2 For the eigenstates of the chiral-helicity the system of corresponding equations can be read (η =, iγ µ µ Ψ η m 1c1+γ 5 Ψ η = 0. (12 h 2 The conjugated eigenstates of the Hamiltonian Ψ + Ψ > and Ψ Ψ > are connected, in fact, by γ 5 transformation Ψ γ 5 Ψ (α ˆpΨ (or m 1 m 1. However, the γ 5 transformation is related to the PT (t t only transformation [4b], which, in its turn, can be interpreted as E E, if one accepts the Stueckelberg idea about antiparticles. We associate Ψ +Ψ > with the positive-energy eigenvalue of the Hamiltonian p 0 = p and Ψ Ψ >, with the negative-energy eigenvalue of the Hamiltonian (p 0 = p. Thus, the free chiral-helicity massless eigenstates may oscillate one to another with the frequency ω = E/ h (as the massive chiral-helicity eigenstates, see [7a] for details. Moreover, a special kind of interaction which is not symmetric with respect to the chiral-helicity states (for instance, if the left chiral-helicity eigenstates interact with the matter only may induce changes in the oscillation frequency, like in the Wolfenstein (MSW formalism. Thus, we derive the same equations as in Ref. [4] beginning with the basic equation (1. The question is: how can these frameworks be connected with the Ryder method of derivation of relativistic wave equations, and with the subsequent analysis of problems of the choice of normalization and that of the choice of phase factors in the papers[7 9]? However, the conclusion may be similar to that which was achieved before: the dynamical properties of the massless particles (e. g., neutrinos and photons may differ from those defined by the well-known Weyl and Maxwell equations [10]. 3 Do not confuse with the Dirac Hamiltonian. 3

5 2. Negative energies in the Dirac equation The recent problems of superluminal neutrinos, e. g., Ref. [11], negative mass-squared neutrinos, various schemes of oscillations including sterile neutrinos, e. g. [12], require much attention. The problem of the lepton mass splitting (e,µ,τ has long history [13]. This suggests that something missed in the foundations of relativistic quantum theories. Modifications seem to be necessary in the Dirac sea concept, and in the even more sophisticated Stueckelberg concept of the backward propagation in time. The Dirac sea concept is intrinsically related to the Pauli principle. However, the Pauli principle is intrinsically connected with the Fermi statistics and the anticommutation relations of fermions. Recently, the concept of the bi-orthonormality has been proposed; the (anti commutation relations and statistics are assumed to be different for neutral particles [8]. We observe some interisting things related to the negative-energy concept. The Dirac equation is: [iγ µ µ m]ψ(x = 0. (13 At least, 3 methods of its derivation exist [2,14,15]: the Dirac one (the Hamiltonian should be linear in / x i, and be compatible with E 2 p p 2 c 2 = m 2 c 4 ; the Sakurai one (based on the equation (E p σ p(e p +σ pφ = m 2 φ; the Ryder one (the relation between 2-spinors at rest is φ R (0 = ±φ L (0, and application of the boosts to them. Usually, everybody uses the following definition of the field operator [16] in the pseudo-euclidean metrics: Ψ(x = 1 d 3 p (2π 3 [u h (pa h (pe ip x +v h (pb h (p]e+ip x ], (14 2E p h as given ab initio. After actions of the Dirac operator at exp( ip µ x µ the4-spinors(u andv satisfythemomentum-spaceequations: (ˆp mu h (p = 0 and (ˆp + mv h (p = 0, respectively; the h is the polarization index. However, it is easy to prove from the characteristic equations Det(ˆp m = (p 2 0 p2 m 2 2 = 0 that the solutions should satisfy the energy-momentum relation p 0 = ±E p = ± p 2 +m 2 in both cases. Let me remind the general scheme of construction of the field operator, which has been presented in [17]. In the case of the (1/2,0 (0,1/2 representation we have: Ψ(x = 1 d 4 pδ(p 2 m 2 e ip x Ψ(p = = = 1 (2π 3 1 (2π 3 h (2π 3 d 4 pδ(p 2 0 Epe 2 ip x u h (p 0,pa h (p 0,p = (15 d 4 p 2E p [δ(p 0 E p +δ(p 0 +E p ][θ(p 0 +θ( p 0 ]e ip x u h (pa h (p = 1 d 4 p [ (2π 3 [δ(p 0 E p +δ(p 0 +E p ] θ(p 0 u h (pa h (pe ip x + 2E h h p + θ(p 0 u h ( pa h ( pe +ip x] = 1 d 3 p [ (2π 3 θ(p 0 u h (pa h (p p0 =E 2E p e i(ept p x + p + u h ( pa h ( p p0 =E p e +i(ept p x] h Duringthecalculationsabovewehadtorepresent1 = θ(p 0 +θ( p 0 inordertogetpositive-and negative-frequency parts. Moreover, during these calculations we did not yet assumed, which 4

6 equation this field operator (namely, the u spinor does satisfy, with negative- or positivemass? In general we should transform u h ( p to the v(p. The procedure is the following one [18]. In the Dirac case we should assume the following relation in the field operator: We know that [15] 4 v h (pb h (p = h h u h ( pa h ( p. (16 ū (µ (pu (λ (p = +mδ µλ, (17 ū (µ (pu (λ ( p = 0, (18 v (µ (pv (λ (p = mδ µλ, (19 v (µ (pu (λ (p = 0, (20 but we need Λ (µ(λ (p = v (µ (pu (λ ( p. By direct calculations, we find mb (µ (p = λ Λ (µ(λ (pa (λ ( p. (21 Hence, Λ (µ(λ = im(σ n (µ(λ, n = p/ p, and b (µ (p = i λ (σ n (µ(λ a (λ ( p. (22 Multiplying (16 by ū (µ ( p we obtain a (µ ( p = i λ (σ n (µ(λ b (λ(p. (23 The equations are self-consistent. In the (1, 0 (0, 1 representation the similar procedure leads to somewhat different situation: a (µ (p = [1 2(S n 2 ] (µ(λ a (λ ( p. (24 This signifies that in order to construct the Sankaranarayanan-Good field operator, which was used by Ahluwalia, Johnson and Goldman [19], it satisfies [γ µν µ ν (i / t E m2 ]Ψ(x = 0, we need additional postulates. For instance, one can try to construct the left- and the right-hand side of the field operator separately each other [20]. However, other ways of thinking are possible. First of all to mention, we have, in fact, u h (E p,p and u h ( E p,p, and v h (E p,p and v h ( E p,p, originally, which may satisfy the equations: 5 [ ] E p (±γ 0 γ p m u h (±E p,p = 0. (25 ( 0 1 Due to the properties U γ 0 U = γ 0, U γ i U = +γ i with the unitary matrix U = = 1 0 γ 0 γ 5 in the Weyl basis, we have [ ] E p γ 0 γ p m U u h ( E p,p = 0. (26 4 (µ and (λ are the polazrization indices here. According to the referee advice I use parenthesis here to stress this. 5 Remember that, as before, we can always make the substitution p p in any of the integrands of (15. 5

7 The properties of the U matrix are opposite to those of P γ 0 P = +γ 0, P γ i P = γ i with the usual P = γ 0, thus giving [ E p γ 0 +γ p m ] Pu h ( E p,p = [ˆp+m]ṽ? (E p,p = 0. While, the relations of the spinors v h (E p,p = γ 5 u h (E p,p are well-known, it seems that the relations of the v spinors of the positive energy to u spinors of the negative energy are frequently forgotten, ṽ? (E p,p = γ 0 u h ( E p,p. Thus, unless the unitary transformations do not change the physical content, we have that the negative-energy spinors γ 5 γ 0 u (see (26 satisfy the accustomed positive-energy Dirac equation. We should then expect the same physical content. Their explicit forms γ 5 γ 0 u are different from the textbook positive-energy Dirac spinors. They are the following ones: 6 ũ(p = N 2m( E p +m p + +m p r p m p r, (27 p l N ũ(p = p +m 2m( E p +m p l. (28 p + m E p = p 2 +m 2 > 0, p 0 = ±E p, p ± = E±p z, p r,l = p x ±ip y. Their normalization is to ( 2N 2. What about the ṽ(p = γ 0 u transformed with the γ 0 matrix? Are they equal to v h (p = γ 5 u h (p? Our answer is no. Obviously, they also do not have well-known forms of the usual v spinors in the Weyl basis, differing by phase factors and in the signs at the mass terms. Next, one can prove that the matrix ( P = e iθ γ 0 = e iθ can be used in the parity operator as well as in the original Weyl basis. However, if we would take the phase factor to be zero we obtain that while u h (p have the eigenvalue +1 of the parity, but (R = (x x,p p (29 PRũ(p = PRγ 5 γ 0 u( E p,p = ũ(p, (30 PRũ(p = PRγ 5 γ 0 u( E p,p = ũ(p. (31 Perhaps, one should choose the phase factor θ = π. Thus, we again confirmed that the relative (particle-antiparticle intrinsic parity has physical significance only. Similar formulations have been presented in Refs. [21], and [22]. The group-theoretical basis for such doubling has been given in the papers by Gelfand, Tsetlin and Sokolik [23], who first presented the theory in the 2-dimensional representation of the inversion group in 1956 (later called as the Bargmann-Wightman-Wigner-type quantum field theory in M. Markov wrote two Dirac equations with the opposite signs at the mass term [21] long ago: [iγ µ µ m]ψ 1 (x = 0, (32 [iγ µ µ +m]ψ 2 (x = 0. (33 In fact, he studied all properties of this relativistic quantum model (while he did not know yet the quantum field theory in Next, he added and subtracted these equations. What did 6 We use tildes because we do not yet know their polarization properties. 6

8 he obtain? iγ µ µ ϕ(x mχ(x = 0, (34 iγ µ µ χ(x mϕ(x = 0. (35 Thus, ϕ and χ solutions can be presented as some superpositions of the Dirac 4-spinors u and v. These equations, of course, can be identified with the equations for the Majorana-like λ and ρ, which were presented in Ref. [7]: 7 iγ µ µ λ S (x mρ A (x = 0, (36 iγ µ µ ρ A (x mλ S (x = 0, (37 iγ µ µ λ A (x+mρ S (x = 0, (38 iγ µ µ ρ S (x+mλ A (x = 0. (39 Neither of them can be regarded as the Dirac equation. However, they can be written in the 8-component form as follows: [iγ µ µ m]ψ (+ (x = 0, (40 [iγ µ µ +m]ψ ( (x = 0, (41 with Ψ (+ (x = ( ρ A ( (x ρ λ S,Ψ (x ( (x = S (x λ A, Γ µ = (x ( 0 γ µ γ µ 0. (42 It is easy to find the corresponding projection operators, and the Feynman-Stueckelberg propagator. You may say that all this is just related to the spin-parity basis rotation (unitary transformations. However, in the previous papers I explained: the connection with the Dirac spinors has been found [7,24]. 8 For instance, λ S (p 1 i 1 i u +1/2 (p λ S (p λ A (p = 1 i 1 i 1 u 1/2 (p 2 1 i 1 i v +1/2 (p, (43 λ A (p i 1 i 1 v 1/2 (p provided that the 4-spinors have the same physical dimension. Thus, we can see that the two 4-spinor systems are connected by the unitary transformations, and this represents itself the rotation of the spin-parity basis. However, it is usually assumed that the λ and ρ spinors describe the neutral particles, meanwhile u and v spinors describe the charged particles. Kirchbach [24] found the amplitudes for neutrinoless double beta decay (00νβ in this scheme. It is obvious from (43 that there are some additional terms comparing with the standard calculations of those amplitudes. One can also re-write the above equations into the two-component forms. Thus, one obtains the Feynman-Gell-Mann [25] equations. As Markov wrote himself, he was expecting new physics from these equations. 7 Of course, the signs at the mass terms depend on, how do we associate the positive- or negative- frequency solutions with λ and ρ. 8 I also acknowledge personal communications from D. V. Ahluwalia on these matters. 7

9 BarutandZiino[22]proposedyetanothermodel. Theyconsideredγ 5 operatorastheoperator of the charge conjugation. Thus, the charge-conjugated Dirac equation has the different sign comparing with the ordinary formulation: [iγ µ µ +m]ψ c BZ = 0, (44 and the so-defined charge conjugation applies to the whole system, fermion + electromagnetic field, e einthecovariantderivative. ThesuperpositionsoftheΨ BZ andψ c BZ alsogiveusthe doubled Dirac equation, as the equations for λ and ρ spinors. The concept of the doubling of the Fock space has been developed in the Ziino works (cf. [23,26] in the framework of the quantum field theory. In their case the self/anti-self charge conjugate states are simultaneously the eigenstates of the chirality. Next, it is interesting to note that we have for the Majorana-like field operators (a η (p = b η (p: [ ] ν ML (x µ +Cν ML (x µ /2 = [( iθφ η η L (pµ 0 d 3 p 1 (2π 3 a η (p µ e ip x + [ ] ν ML (x µ Cν ML (x µ /2 = [( η 0 φ η L (pµ a η (p µ e ip x + (45 2E p ( 0 a η(p ] φ η µ e ip x, L (pµ d 3 p 1 (2π 3 (46 2E p ( iθφ η (pµ L a η(p ] µ e ip x, 0 which, thus, naturally lead to the Ziino-Barut scheme of massive chiral fields, Ref. [22]. Finally, I would like to mention that, in general, in the Weyl basis the γ matrices are not Hermitian, γ µ = γ 0 γ µ γ 0. So, γ i = γ i, i = 1,2,3, the pseudo-hermitian matrix. The energymomentum operator i µ is obviously Hermitian. So, the question, if the eigenvalues of the Dirac operator iγ µ µ (the mass, in fact would be always real? The question of the complete system of the eigenvectors of the non-hermitian operator deserve careful consideration [27]. Bogoliubov and Shirkov [17, p.55-56] used the scheme to construct the complete set of solutions of the relativistic equations, fixing the sign of p 0 = +E p. The main points of this Section are: there are negative-energy solutions in that is previously considered as positive-energy solutions of relativistic wave equations, and vice versa. Their explicit forms have been presented in the case of spin-1/2. Next, the relations to the previous works have been found. For instance, the doubling of the Fock space and the corresponding solutions of the Dirac equation obtained additional mathematical bases. Similar conclusion can be deduced for the higher-spin equations. 3. Non-commutativity in the Dirac equation. The non-commutativity [28, 29] exibits interesting peculiarities in the Dirac case. We analized Sakurai-van der Waerden method of derivations of the Dirac(and higher-spins too equation[30]: (EI (4 +α p+mβ(ei (4 α p mβψ (4 = 0. (47 Obviously, the inverse operators of the Dirac operators of the positive- and negative- masses exist in the non-commutative case. As in the original Dirac work, we have β 2 = 1, α i β +βα i = 0, α i α j +α j α i = 2δ ij. (48 8

10 For instance, their explicit forms can be chosen ( α i σ i 0 = 0 σ i, β = ( ( We also postulate the non-commutativity relations for the components of 4-momenta: [E,p i ] = Θ 0i = θ i, (50 as usual. Therefore the equation (47 will not lead to the well-known equation E 2 p 2 = m 2. Instead, we have { } E 2 E(α p+(α pe p 2 m 2 i(σ I (2 [p p] Ψ (4 = 0 (51 For the sake of simplicity, we may assume the last term to be zero. However, it may be used for discussion of extra physical generations. Thus, we come to { } E 2 p 2 m 2 (α θ Ψ (4 = 0. (52 However, let us apply the unitary transformation. It is known [7,31] that one can 9 For α matrices we re-write (53 to U 1 (σ au 1 1 = σ 3 a. ( U 1 (α θu1 1 = θ = α 3 θ. ( The explicit form of the U 1 matrix is (a r,l = a 1 ±ia 2, a = a : U 1 = ( 1 a+a3 a l = 2a(a+a3 a r a+a 3 1 2a(a+a3 [a+a 3 +iσ 2 a 1 iσ 1 a 2 ], ( U1 0 U 1 =. (55 0 U 1 Let us apply the second unitary transformation: U 2 α 3 U 2 = α = ( The final equation is [E 2 p 2 m 2 γchiral θ ]Ψ 5 (4 = 0. (57 In the physical sense this implies the mass splitting for a Dirac particle over the non-commutative space, m 1,2 = ± m 2 ±θ. This procedure may be attractive for explanation of the mass creation and the mass splitting for fermions, because we can associate m = m 1 m 2 to the mass difference of electron and muon, for instance. 9 Some relations for the components of an arbitrary vector a should be assumed. Moreover, in our case θ should not depend on E and p. Otherwise, we must take the non-commutativity [E,p i ] into account again. 9

11 4. Conclusions In this contribution we considered the generalized equation in the (1/2, 0 (0, 1/2 representation. It is different from the well-known Weyl equation in the massless limit. It leads to different physical consequences. Next, we showed that the negative-energy solutions are possible in the Dirac equation. Finally, we showed that the non-commutativity parameter may give us possibility to describe fermionic mass splitting. Acknowledgments This talk was also presented at the 31st ICGTMP, June 19-25, 2016, CBPF, Rio de Janeiro, Brasil. I appreciate useful discussions with colleagues. References. [1] Gersten A 1999 Found. Phys. Lett Gersten A [2] Sakurai J J 1967 Advanced Quantum Mechanics (Addison-Wesley 3.2. [3] Dvoeglazov V V 2000 J. Phys A: Math. Gen [4] Tokuoka Z 1967 Prog. Theor. Phys Gupta N D S 1967 Nucl. Phys. B4 147 Santhanam T S and Chandrasekaran P S 1969 Prog. Theor. Phys Fushchich V I 1970 Nucl. Phys. B Fushchich V I 1972 Lett. Nuovo Cim Fushchich V I and Grischenko A 1970 Lett. Nuovo Cim Simon M T 1971 Lett. Nuovo Cim Santhanam T S and Tekumalla A R 1972 Lett. Nuovo Cim [5] Raspini A 1994 Int. J. Theor. Phys Raspini A 1996 Fizika B Raspini A 1997 Fizika B Raspini A 1998 Fizika B 7 83 [6] A. Raspini 1999 in Photon and Poincaré Group. Dvoeglazov VV (ed. pp [7] Dvoeglazov V V 1995 Int J. Theor. Phys Dvoeglazov V V Dvoeglazov V V 1996 Nuovo Cim. B Dvoeglazov V V 1995 Nuovo Cim. A Dvoeglazov V V 1997 Hadronic J Dvoeglazov V V 1997 Fizika B6 111 Dvoeglazov V V 1997 Adv. Appl. Clifford Algebras 7(C 303 Dvoeglazov V V 1999 Adv. Appl. Clifford Algebras9 231 Dvoeglazov V V 1998 Acta Physica Polon. B Dvoeglazov V V 2000 Found. Phys. Lett [8] Ahluwalia D V 1996 Int. J. Mod. Phys. A Ahluwalia D V 1998 Mod. Phys. Lett. A [9] Dvoeglazov V V 2000 Czech. J. Phys in Photon: Old Problems in Light of New Ideas. Dvoeglazov V V (ed. pp [10] Dvoeglazov V V 2016 Adv. Appl. Clifford Algebras Online First (2016 [11] Guang-jiong Ni 2004 in Relativity, Gravitation, Cosmology (Nova Science Pub., Huntington, NY, USA, 2004 (Preprint hep-ph/ [12] Bilenky S M and Giunti C 1996 Nucl. Phys. Proc. Suppl [13] Barut a O 1978 Phys. Lett. B Barut A O 1979 Phys. Rev. Lett [14] Dirac P A M 1928 Proc. Roy. Soc. Lond. A [15] Ryder L H 1985 Quantum Field Theory. (Cambridge University Press, Cambridge [16] Itzykson C and Zuber J-B Quantum Field Theory. (McGraw-Hill Book Co. p. 156 [17] Bogoliubov N N and Shirkov D V 1973 Introduction to the Theory of Quantized Fields. 2nd Edition. (Nauka, Moscow [18] Dvoeglazov V V 2003 Hadronic J. Suppl (Preprint physics/ Dvoeglazov V V 2006 Int. J. Mod. Phys. B

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